From Wikipedia, the free encyclopedia.
In general geodesic stay for the curves which are "straight" in a sense.
Geodesic is a curve which is locally distance minimizer. More precicely if is a metric space a curve is a geodesic if there is a constant such that for any there is a neighborhood of in such that for any we have .
If the last equality is satisfyed on all it is called minimizing geodesic or shortest path
The most familiar examples are the straight lines in Euclidean geometry.
On a sphere, the geodesics are the great circles.
The shortest path from point A to point B on a sphere is given by the shorter piece of the great circle passing through A and B. Note that if A and B are antipodal points (like the North pole and the South pole), then there are many shortest paths between them. In general, metric space can have no geodesic exapt constant curves.
If is Riemannian manifold then geodesics are allways smooth curves
so one can define , and the above definition is equivalent to ,
where stays for covariant derivative.
The last definition has sense for all manifolds with connection in particular for
Levi Civita connection on
Pseudo-Riemannian manifolds.
The space-time in the theory of general relativity is a Pseudo-Riemannian manifold, and geodesic can be defined exactly as before. In Space-time, particles travel along geodesics. Everything in "free fall" such as the orbit of an astronaut, or the orbit of a planet follows a so called timelike geodesic, also called a world line. Light (photons in general) follow a path called nul geodesics.For metric spaces
Riemannian and pseudo-Riemannian manifolds
General relativity
